Optimal Paths in Real Multimodal Transportation Networks: an Appraisal Using GIS Data from New Zealand and Europe

Proceedings of the 45th Annual Conference of the ORSNZ, November 2010 Optimal Paths in Real Multimodal Transportation Networks: An Appraisal Using GIS Data from New Zealand and Europe Felipe Lillo, Frederik Schmidt School of Computing and Mathematical Sciences AUT University Auckland, New Zealand {flillovi,fschmidt}@aut.ac.nz Abstract The most convenient route to connect two locations is often a mix of different transportation systems. For instance, a user can make intercity trips by selecting a combination of transport modes such as car, rail, ship or airplane. In this case the transportation system is said to be multimodal. In this paper, real multimodal transportation systems are experimentally analyzed. Real road–rail networks from Denmark, Hungary, Spain, Norway and New Zealand are built based on a set of digitized transportation maps obtained from several GIS libraries. These networks are modelled as coloured–edge graphs to be used as main input by a multimodal Dijkstra’s algorithm that computes a set of optimal paths. The cardinality of the resulting set is at the core of the approach tractability. It is concluded that vertices connectivity and network shape considerably affect the total number of optimal paths. Key words: Multimodal network, coloured–edge graph, GIS libraries, transportation networks. 1 Introduction A multimodal network (MMN) is a transportation system that considers two or more transport choices for connecting locations (vertices) in a network. Freight and urban transportation are two application fields in which multimodal networks are extensively found. In (Bontekoning, Macharis, and Trip 2004) freight transportation is thoroughly reviewed for identifying possible research trends. The authors here conclude research in multimodal freight transportation is still in a pre-paradigmatic phase. Urban transportation deals with the movement of passengers in urban areas considering variables such as congestion levels (flow), public fares, transport modes (bus, subway, private car or bicycle), service demand and user behaviour. Some 281 Proceedings of the 45th Annual Conference of the ORSNZ, November 2010 overviews about urban transportation are found in (Boyce 2007), (Lee 1994), (We- gener 1994) and (Nagurney 1984). Despite their popularity as multimodal research field, freight and urban transportation are not the only areas where multimodal networks can be used as a modelling tool. Application studies into computer networks (Nigay and Coutaz 1993), biomedicine (Heath and Sioson 2007) and manufacturing (Medeiros et al. 2000) can be noticed in the literature too. One problem associated to MMN is the determination of a shortest path from an origin to a destination. The shortest path problem has been extensively studied in both practice and theory. Likewise, a large number of algorithms have been devel- oped by different fields of enquire such as operation research and computing sciences. When dealing with algorithms for multimodal networks, researchers and practition- ers typically opt for techniques based on variations of classical approaches such as Dijkstra’s algorithm or Bellman–Ford method (a description of these approaches is found in (Lawler 2001)). However, a direct application of these approaches generates a shortest path that does not consider the multimodal traits of the network as part of the analysis. Thereby, an optimal combination of means of transport is an outcome of the shortest path itself. Consequently, a multimodal network cannot be treated as a “unimodal” because each edge includes an additional variable (the mode) that has to be included as part of the analysis. An appraisal is carried out in this paper to analyze the comportment of the shortest path problem in real MMNs. The employed modelling approach takes a coloured–edge graph that represents modes, cities and intercity links by colours, vertices and edges. This graph is used by a generalization of Dijkstra’s algorithm that computes a set of optimal paths. This set of paths is the key of the model’s tractability so that experiments are set for tracing its value. Unlike others approaches, the presented modelling and computational techniques are able to keep the multimodal traits of the network throughout the analysis. The results indicate that the total number of optimal paths is significantly influenced by the shape and connectivity of the network. The remainder of this paper is organized in four further sections. First the modelling approach and the algorithm are described in section 2. The experimental setup is described in section 3. Results are given in section 4. Finally, section 5 provides conclusions and future work in this research field. 2 Model and Algorithm 2.1 The Coloured–Edge Graph Model The coloured–edge graph is a modelling tool introduced by (Ensor and Lillo 2009) for the modelling of MMNs. This graph modelling approach labels edges with colours for representing a specific attribute such as a transport mode. In a real transportation system, two locations might be connected by several modes. For this case, the coloured–edge graph allows the use of multiple edges. These edges can be directed or undirected as well as weighted or unweighted. In this research weighted coloured– edge digraphs are utilized to model real multimodal transportation networks from several countries. In their work (Ensor and Lillo 2009) formally define a weighted coloured–edge graph as G = (V,E,ω,λ) which is a directed graph (V, E) with vertex set V and edge set E, a weight function ω: E → R+ on edges, and a colour function λ: E → M 282 Proceedings of the 45th Annual Conference of the ORSNZ, November 2010 on edges. M is a finite set of colours with k = |M|. Each edge euv ∈ E joining vertices u and v has a positive weight ω(euv) and a colour λ(euv). For any colour i ∈ M and for any path puv between two vertices u and v, the path weight ωi(puv) i ω p ω e in colour is defined as i( uv) = euv∈p,λ(euv)=i ( uv). The total path weight is represented as a k-tuple (ω (e ),...,ω (e ),...,ω (e )), giving the total weight 1 uv P i uv k uv of the path in each colour. Only limited research has been conducted in the area of coloured–edge graphs. (Climaco, Captivo, and Pascoal 2010) studied the number of spanning trees in a weighted graph whose edges are labelled with a colour. This work defines weight and colour as two conflicting criteria. Hence, the proposed algorithm generates a set of nondominated spanning trees. The computation of coloured paths in a weighted coloured–edge graph is investigated by (Xu et al. 2009). The main feature of their approach is a graph reduction technique based on a priority rule. This rule basically transforms a weighted coloured–edge multidigraph into a coloured–vertex digraph by applying algebraic operations upon the adjacent matrix. Additionally, (Xu et al. 2009) provide an algorithm to identify coloured source–destination paths. Nevertheless, the algorithm is not intended for general instances because it just works with the number of edges as a path weight. Furthermore, only paths having consecutive edges with distinct colours are considered. 2.2 Multimodal Dijkstra’s Algorithm A weighted coloured–edge graph can produce a factorial number of source–destination paths. For instance, the total number of paths in a complete coloured–edge graph is O (kn−1(n − 2)!). This can be proved by applying a basic counting argument. However, the question is how many of these paths are optimal (shortest). At first glance, traditional shortest path algorithms might be able to provide an answer. Nevertheless, these procedures were originally designed for “unimodal” networks. Thus, a direct application of such algorithms on a coloured–edge graph produces outcomes that do not take transport modes into consideration. This section explain a procedure designed by (Ensor and Lillo 2009) that extracts a set of optimal path from a weighted coloured–edge graph base on a general version of the well–known Dijkstra’s algorithm. Firstly, some notation need to be introduced. Let Muv be the set of shortest paths joining two vertices u and v in a weighted coloured–edge graph. This set can be extracted from a weighted coloured–edge graph by defining a partial order relation on weight tuples. As a result, a smaller set of paths is obtained. Each k–tuple in Muv is Pareto efficient which means that a weight in a tuple cannot be improved (or worsened) without worsening (or improving) the same weight of another. The cardinality of Muv is a key factor in determining the tractability of the model. The computation of Muv is performed in (Ensor and Lillo 2009) by a generalization of Dijkstra’s algorithm. Unlike its classic counterpart, the multimodal Dijkstra’s algorithm (MDA) has a partially ordered data structure to manage path weights. The algorithm takes as input a coloured–edge network G and a source vertex s. It commences at s with the empty path and relaxes each edge that is incident from the source vertex s, adding the single edge paths to the queue. At the front of the queue will be a shortest path estimate to some vertex v adjacent to s. Next, the algorithm determines the shortest paths to v (note that there are more than one) and relaxes edges incidents to v. This iterative process finishes when the queue has 283 Proceedings of the 45th Annual Conference of the ORSNZ, November 2010 no more paths to be compared. 3 Experimental Setup The experimental study collected vector data information about Denmark, Hun- gary, Spain, Norway and New Zealand from a GIS library (Geofabrik 2010). These countries were selected based on their similarities in shape and number of locations. For example, New Zealand has resemblances with Norway in shape and number of vertices.

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